Normalized defining polynomial
\( x^{46} - 2 x + 3 \)
Invariants
| Degree: | $46$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 23]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-749085428928598873917583984529112970694334716941605800990955391911147726327162798519362537914368=-\,2^{47}\cdot 3^{45}\cdot 40739891922760603\cdot 108253239392882437561\cdot 408511528412100333650099\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $121.40$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 40739891922760603, 108253239392882437561, 408511528412100333650099$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $\frac{1}{11} a^{45} + \frac{3}{11} a^{44} - \frac{2}{11} a^{43} + \frac{5}{11} a^{42} + \frac{4}{11} a^{41} + \frac{1}{11} a^{40} + \frac{3}{11} a^{39} - \frac{2}{11} a^{38} + \frac{5}{11} a^{37} + \frac{4}{11} a^{36} + \frac{1}{11} a^{35} + \frac{3}{11} a^{34} - \frac{2}{11} a^{33} + \frac{5}{11} a^{32} + \frac{4}{11} a^{31} + \frac{1}{11} a^{30} + \frac{3}{11} a^{29} - \frac{2}{11} a^{28} + \frac{5}{11} a^{27} + \frac{4}{11} a^{26} + \frac{1}{11} a^{25} + \frac{3}{11} a^{24} - \frac{2}{11} a^{23} + \frac{5}{11} a^{22} + \frac{4}{11} a^{21} + \frac{1}{11} a^{20} + \frac{3}{11} a^{19} - \frac{2}{11} a^{18} + \frac{5}{11} a^{17} + \frac{4}{11} a^{16} + \frac{1}{11} a^{15} + \frac{3}{11} a^{14} - \frac{2}{11} a^{13} + \frac{5}{11} a^{12} + \frac{4}{11} a^{11} + \frac{1}{11} a^{10} + \frac{3}{11} a^{9} - \frac{2}{11} a^{8} + \frac{5}{11} a^{7} + \frac{4}{11} a^{6} + \frac{1}{11} a^{5} + \frac{3}{11} a^{4} - \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{4}{11} a - \frac{1}{11}$
Class group and class number
Not computed
Unit group
| Rank: | $22$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{46}$ (as 46T56):
| A non-solvable group of order 5502622159812088949850305428800254892961651752960000000000 |
| The 105558 conjugacy class representatives for $S_{46}$ are not computed |
| Character table for $S_{46}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $22{,}\,18{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ | $32{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ | $19{,}\,16{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | $25{,}\,{\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $26{,}\,16{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $15{,}\,{\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | $41{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | $16{,}\,{\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | $36{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | $44{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | $24{,}\,21{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $25{,}\,15{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ | $19^{2}{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $31{,}\,{\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 40739891922760603 | Data not computed | ||||||
| 108253239392882437561 | Data not computed | ||||||
| 408511528412100333650099 | Data not computed | ||||||